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Macromolecules 1984, 17, 2674-2678
Rotational Isomeric State Calculations of the Dynamic Structure Factor and Related Properties of Some Linear Chains. 2. First Cumulant of the Dynamic Structure Factor Kanji Kajiwarat and Walther Burchard** Institute for Chemical Research, Kyoto University, Uji, Kyoto-fu, 611 Japan, and Znstitut fur Makromolekulare Chemie der Universitat Freiburg, 7800 Freiburg i. Brsg., West Germany. Received March 16, 1984 ABSTRACT The particle scattering factor and the first cumulant of the dynamic structure factor of poly(methylene),poly(oxymethylene),cis-l,4-polybutadiene,and trans-1,4-polybutadienewere calculated by the same technique as outlined in part 1. For q > 0.2 nm-l, which corresponds to a length of the subchain of rij < 0.8 nm, the influence of directional correlation due to chain stiffness or rotational hindrance becomes effective and causes the expected deviations from the Gaussian behavior. As expected,no general form for the asymptote of r in this far-q region was obtained for the nondraining limit. This region is defined by the special characteristicsof the microstructure of the individual chains. When the free-drainingcontribution is taken into account, the reduced first cumulant r/q2in a certain region increases linearly with q. The proportionality coefficient increases slightly with the characteristic ratio C,, i.e., with chain stiffness, and is in every case larger than that of a Gaussian chain. This coefficient in the asymptotic region of large q is related to the coefficient C in r / q 2= D(l + C ( S 2 ) q 2- ...) of the reduced first cumulant at small q values and again for the simulated chains is larger than for the Gaussian chain. Introduction In the preceding paper, part 1,' of this study we reported rotational isomeric state (RIS) calculations of the geometric and hydrodynamic radii and of the related p parameters for a number of realistic chain molecules. A slight increase of the p parameter with the characteristic ratio C,, Le., chain stiffness, was found and discussed. The present paper now deals with the particle scattering factor P ( q ) and the angular dependence of the first cumulant r of the dynamic structure factor S(q,t) for the same chains as those treated in part 1. Again the RIS model has been chosen as the basis of the calculations. According to Akcasu and Guro1,2the first cumulant of the dynamic structure factor is given in terms of equilibrium averages as follows:
(exp(iqrjk)),, = 2-l sin z
(3)
((qDjk.9) eXP(iq.r;k)) = q2kBT[6jk/{j + ((16jk)/(4~q0rjk))(~-1 sin z + z-2 cos z - r3sin z ) ] (4) with 2
= qrjk
(5)
In the second step the averaging is performed over all distances r;k. For linear Gaussian chains an analytical expression can be obtained. Neglecting the free-draining term and replacing the double sum of the nondraining part by an integral, eq 1 with eq 3 yields in the nondraining limit r / q 2 = (3q2uA/4N1/2)X [u + D(u) + 2(u2 - 1)G(u)]/[u2- 1 + exp(-u2)] (6) where
where the angular brackets denote the average over the equilibrium distance distribution between the two chain elements j and k. Djk is the element of the diffusion tensor for this pair of chain elements and is given by
Here, rjkis the vector from the j t h to the kth element, qo the solvent viscosity, and S;the frictional coefficient of the jth element. The first term results from the friction of the individual bead, while the second term takes account of the hydrodynamic interaction which the kth chain element in the chain exerts on the j t h element. Equation 1implies that the total hydrodynamic interaction in the chain can be approximated by the sum over all chain element pairs j # k . The denominator in eq 1 is identified as the static structure factor S(q) = P P ( q ) , where P(q) is the particle scattering factor. The averaging procedure may be carried out in two steps. When the vector r;k. is averaged over all orientations, then the scattering function for a pair of chain elements is converted into28
' Institute for Chemical Research, Kyoto University.
* Institut fur Makromolekulare Chemie der Universitat Freiburg. 0024-9297/84/2217-2674$01.50/0
u2 = q 2 ( S 2 )
(7)
D ( x ) = exp(-xz)Ixexp(t2) 0 dt
(9)
A = kgT/(61/2~3/2bvo)
(10)
with (S2) being the mean square radius of gyration and b the effective bond length. The first cumulant of eq 6 has a characteristic asymptote at large q
-
( r / q 2 ) (k~T/1670)q
(11)
which is independent of the chain length, or the mean square radius of gyration, and independent of polymer polydispersity. The result of eq 11 is, however, limited to Gaussian chain statistics and to qb > 1but is valid for any type of polymer architecture, i.e., cyclization or branching. The region of small and intermediate q values depends significantly, of course, on the polymer architecture. For small q, i.e., q2(S2) 1nm-', we fixed N = 400 for each polymer in the present simulation.
Results a n d Discussion Particle Scattering Factor. The particle scattering factor for small q 2 ( S 2 )is fully determined by the mean square radius of gyration and in this region shows no difference for the various polymers when plotted against q2(S2)= u2. Differences in the shape become observable at larger u2 and in particular in the asymptotic region of u2 >> 1. An appropriate means of studying the asymptotic region is the Kratky plot, where q2P(q) is plotted against q. For Gaussian chains the particle scattering factor in this plot approaches a plateau, whereas a thin rodlike molecule exhibits a straight line with a positive slope. Because of the hindrance to internal rotation, every realistic chain will generate a specific microstructure which, in many cases, can be approximated by a rod extending over a certain length, which is called the persistence length. According to Porod and Kratky the change (now usually called the "crossover") from the Gaussian to the rodlike behavior should occur at qu = 1.91?l0 where a is the persistence length, which for the flexible chains of the present consideration is on the order of 1 nm. Figure 1 shows the Kratky plots for the simulated poly(methylene), poly(oxymethylene), and cis-1,4- and trum-l,4-polybutadienes.Zierenberg et al? have employed the same method to estimate the particle scattering factor for poly(methylene),and their result is consistent with the present findings. Deviations from the Gaussian chain behavior are observed in all cases when q exceeds 0.2 nm-', which is by a factor 5 earlier than expected from Porods estimation. Similar deviations have been found also for the idealized wormlike chain, where the crossover never appears as a kink point but is actually a broader region which starts already at much lower q values. The scattering behavior at large q > 1 nm-l is expected to show the rodlike behavior for the wormlike chain but will show special characteristics for realistic chains as result of their individual microstructure. A wobbled helix, for instance, as realized by poly(oxymethylene), will show different behavior from that of a cis-1,4-polybutadiene chain. Hence no general asymptotic behavior can be expected in accordance with the present findings. Recently,
Macromolecules, Vol. 17, No. 12, 1984
2676 Kajiwara and Burchard
I
p, ,
i
,
,
10
20
10
30"
1
%/ I
*O
u
30
Figure 5. Normalized apparent diffusion coefficient of trans1,4-polybutadiene(symbols as in Figure 2). Table I Coefficient C in Eq 12, Characteristic Ratio C,,Parameter p, and Value of the Asymptote r at Large q after Monte Carlo Simulations on the Basis of the RIS Model" I
P1 10-
.*/, ,,,, ,,,,,,,...... "".."""".... ....., c)(,
--/
-,?
..'
.
,
.-...
POLYOXYMETHYLENE
I
poly(methy1ene) poly(oxymethy1ene)
0.1804 0.1851 0.1870 0.2022
C. 3.76 5.80 6.87 8.16
1.45 1.46 1.48 1.60
0.067 0.070 0.072 0.078
Gaussian chain
0.1733
1.00
1.504
0.0625
trans-1,4-polybutadiene
/
t
C
polymer cis-l,4-polybutadiene
i-/
5-
(?o/kBT)*
1
/
I
l
polystyrene
l
Figure 3. Normalized apparent diffusion coefficient of poly(oxymethylene) (symbols as in Figure 2).
=.
A*>-
10
*O
Experimental Data poly(dimethylsi1oxane) 5.7-6.2 poly(methy1 methacrylate) 6.9
u
Figure 4. Normalized apparent diffusion coefficient of cis-1,4polybutadiene (symbols as in Figure 2). Edwards et a1.16 have carried out similar Monte Carlo simulations on the basis of the RIS model for the particle scattering factor of cyclic poly(dimethylsi1oxanes). Their technique seems to differ, however, from that used in this paper. The apparent diffusion coefficient, D,,,,, or reduced first cumulant, I?/$ = Dapp(q),behaves also characteristically for each polymer as is shown in Figures 2-5. The behavior at small q, i.e., u2 400
RIS Calculations of the Dynamic Structure Factor. 2 2677
Macromolecules, Vol. 17, No. 12, 1984 10-l
0.20
8 6
/
0
L
.
1.0
x
1 I
2t 0161
2t+2
L.
2 - 2 10
2
4
5
6
7
1.4
I
,
3
a
l4
:/ 4 6 810.'
2
I
t 12c
4 6 810'
q/nm-'
Figure 6. log-log plots of the a parent diffusion coefficients against q, where A = kBT!(6s3/{,): (a) poly(methy1ene);(b) poly(oxymethy1ene); (c) cu-1,4-polybutadiene; (d) trans-1,4polybutadiene. (The free draining term is taken into consideration.) Deviations from the intermediate asymptotes occur at lower q values for the stiffer chains (b) and (d).
remains indeed constant for the arguments which were presented in part 1 (no change of p and C, when N is increased). At large q values the predicted asymptotic behavior of r/q3= constant was not attained for any of the chosen chains when nondraining is assumed. Such behavior can be expected only as long as the free-draining contribution can be neglected, Le., in a region of qb < 0.1. A t larger q values, which correspond to subchain length of N N 10, the free-draining term has already a remarkable influence. Taking into consideration the free-draining term rfree draining
= (kBT/n(q2/NP(q))
(17)
we find, indeed, over a wide range the expected q3 asymptote with values for ( q o / k B r ) ( r / q 3as ) given in the last column of Table I. Here the frictional coefficient was assumed as C = 6a~o(b/2) (18) which corresponds to B = 0.69 in Akcasu's notation.12 Our results are qualitatively in agreement with earlier calculations by Akcasu et al.,13 who also took into account the free-draining contribution but who considered only Gaussian chains. For the more realistic chains the values for the asymptotes seem in this simulation slightly to increase with the characteristic ratio C, as is shown in Table I. With increasing chain stiffness the region of a q3 dependence apparently becomes smaller, and deviations toward a weaker asymptotic behavior occur already at lower q values. This is shown for the poly(oxymethy1ene) chain in (Figure 6d). Nicholsen et al.14have observed a similar tendency with various polymers by quasi-elastic neutron scattering. Poly(dimethylsi1oxane) as the most flexible chain shows, for instance, the expected q3 dependence for r while polystyrene and poly(tetrahydrofuran)with larger persistence lengths exhibit lower exponents than 3 in their q dependences. Eventually, for very large q , in a region where q b >> 1, a q2 dependence of r is again observed, which reflects the local chain relaxation. The region of the transition from the q3 dependence of a quasi-Gaussian chain to the final q2 dependence is certainly characteristic of the local chain structure but has not sufficiently been
Cm
Figure. 7. Dependence of the initial coefficient C in the angular dependence of the apparent diffusion coefficient on the characteristic ratio (chain stiffness) C, (upper part) and the dependence of the p parameter on the characteristic ratio C,; the open circles represent the data of the computer simulation and the filled circles represent the experimentally observed data (see Table I). Curves indicated by 1.5 and 3 A indicate the chain diameters in the hybrid theory of Koyama.
covered by the present simulation. No general behavior can be expected and indeed is observed. We close this discussion with two remarks. Benmouna et al.15 predict on the basis of their sliding-rod model two intermediat subasymptotes. One corresponds to the q3 dependence as already discussed; the second should show a q3 In ( l l q b ) behavior of a rigid rod at larger q values. We have observed no noticeable crossover to a stronger than q3 dependence. Possibly the simulated chains are too short as to warrant the asymptotic behavior predicted by the sliding-rod model. The sliding-rod model gives, apparently: a good description of the dynamic structure factor at small q. The second remark is concerned with the coupling of translational and rotational motions. Such coupling is probably small for Gaussian chains but should not be dismissed for rigid rods. Recent calculations by Rallison and Lea15t6show, for instance, for the coefficient C in eq 12 a drop of l/lo when the coupling is neglected (Pecora) when the translation-rotation coupling is down to taken into account. This effect may have a significant influence on the asymptotic region of very large q and requires a modification of the present model. This modification will be treated in a forthcoming paper. In summary, we can state the following: (i) The application of the RIS model, as the better representation of a real chain next to the Gaussian chain model, shows dependence of the p parameter, the coefficient C, and the coefficient in the q3 dependence of r on chain stiffness. (ii) These results stand in qualitative good agreement to a recent theory by Schmidt and Stockmayer developed on the basis of the wormlike chain model! (iii) The far-q region requires a modification of the present simulation to take into account the coupling between translation and rotation of the rodlike segments.
References and Notes (1) Kajiwara, K.; Burchard, W. Macromolecules, preceding paper in this issue. (2) Akcasu, A. Z.; Gurol, H. J.Polym. Sci., Polym. Phys. Ed. 1976, 14, 1.
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Macromolecules 1984,17, 2678-2685
(3) Burchard, W.; Schmidt, M.; Stockmayer, W. H. Macromolecules 1980, 13, 580. (4) Burchard. W. Adv. Polvm. Sci. 1983. 48. 1. (5) Schmidt, .M.; Stockmaier, W. H. Mucromolecules 1984, 17, 509. (6) Rallison, J. M.; Leal, L. J. Chem. Phys. 1981, 74, 4819. (7) Flory, P. J. "Statistical Mechanics of Chain Molecules"; In-
terscience: New York, 1969. (8) Zierenberg, B.; Carpenter, D. K.; Hsieh, J. H. J . Polym. Sci., Polym. Symp. 1976, No. 54, 145. (9) Kratky, 0.; Porod, G . Recl. Trau. Chim. Pays-Bas 1949, 68, 1106.
(10) Koyama, R. J . Phys. SOC.Jpn. 1973,34, 1029. (11) Schmidt, M., Personal communication. (12) Akcasu, A. Z.; Benmouna, M.; Han, C. C. Polymer 1980, 21, 866. (13) . . Benmouna. M.: Akcasu. A. Z. Macromolecules 1980. 13. 409. (14) Nicholson, L. K.;Higgins, J. S.; Hayter, J. B. Macromolecules 1981, 14, 836. (15) Benmouna, M.; Akcasu, A. Z.; Daoud, M. Macromolecules 1980, 13, 1703. (16) Edwards, C. J. C.; Richards, R. W.; Stepto, R. F. T.; Dodgson, K.; Higgins, J. S.; Semlyen,J. A. Polymer 1984, 25, 365. I
,
Liquid-Crystalline Ordering in Solutions of Semiflexible Macromolecules with Rotational-Isomeric Flexibility Alexei R. Khokhlov* and Alexander N. Semenov Physics Department, Moscow State University, Moscow 11 7234, USSR. Received December 28, 1983
ABSTRACT The orientational ordering in solutions of semiflexiblemacromoleculeswith rotational-isomeric flexibility is considered. It is shown that in this case the liquid-crystallinetransition has many features essentially different from the corresponding features for persistence or freely jointed flexibility models: the isotropic phase becomes unstable at lower polymer concentrations and the appearing anisotropic phase is much more ordered and more concentrated; the width of the phase separation region is thus anomalously large. The transition properties depend essentially on the ratio of two flexibility parameters: the length of the usual Kuhn segment of the chain controlled by the rotational isomerism and the length of a second Kuhn segment (concept introduced in this paper) connected with a small persistent flexibility component, which is always present in real chains. The behavior of a solution of semiflexible rotational-isomeric chains in external orientational fields of dipole and quadrupole types is analyzed as well. It is shown that to suppress the phase transition, which takes place when the solution is concentrated, it is necessary to apply much stronger fields than for the persistence or freely jointed flexibility models. A phase transition can be induced by an external orientational field at practically any concentration of the initially isotropic solution. The susceptibility of the solution in zero dipole field, xo,in the case under consideration increases with the solution concentration in the anisotropic phase, c, accordingto the power law xo c6/* (to be compared with the exponential increase for the persistence model and with practical independence of c for the freely jointed model).
-
1. Introduction It is well-known that in sufficiently concentrated solutions of stiff-chain polymers an orientationally ordered phase can appear. In ref 1-5 the corresponding liquidcrystalline phase transition was studied in detail for the case of completely stiff macromolecules (long, absolutely rigid rods). Accounting for the partial flexibility of the chain of a stiff polymer naturally constitutes the next fundamental problem. This problem has been considered in recent papers by Flory and Matheson6J and by Ronca and Yoon,22-24and in our papers.8-11 From the theoretical point of view the simplest partially flexible chains are the so-called semiflexible macromolecules whose effective Kuhn segment length, 1, is much smaller than the total contour chain length, L, but much larger then the width of the chain, d L >> 1 >> d. The consideration of a solution of chains for which L 1 is much more complex." Semiflexible macromolecules can differ depending on the flexibility mechanism of a polymer chain. Simplest is the freely jointed model, where the chain can be represented as a sequence of freely jointed rigid rods of length 1 and diameter d (Figure 1). Orientational ordering in solutions of semiflexiblefreely jointed macromolecules was considered in 1978 independently in ref 6 and 8. It was shown that the properties of the liquid-crystalline transition are practically the same as in a solution of disconnected rigid rods of length 1 and diameter d. It should be emphasized, however, that freely jointed chains are encountered very rarely in the real polymer world. The most common flexibility models are the per-
-
0024-9297/84/2217-2678$01.50/0
sistence one (which is due to the gradual accumulation of the effect of small thermal vibrations of valence angles, bonds, etc.) and the rotational-isomeric one.12 Persistent semiflexible chains can be represented as long, absolutely homogeneous elastic filaments of width d and a Kuhn segment length 1 (Figure 2). The liquid-crystalline transition in a solution of semiflexible persistent chains was considered in ref 10; it was shown that it takes place at significantly higher polymer concentrations and that the order parameter at the transition point is much smaller than for a solution of freely jointed chains with the same parameters d and 1. Thus, the results of ref 10 permit us to conclude that the conditions for orientational ordering in solutions of semiflexible polymers depend in an essential way on the mechanism of the chain flexibility. It is consequently important to analyze the liquidcrystalline transition in solutions of semiflexible chains with other flexibility models (differing from the freely jointed and persistence ones) and, primarily, with the widely used rotational-isomeric flexibility model. This is precisely the aim of the present paper. It is noteworthy that the majority of known polymers able to form a liquid-crystalline phase in solution (helical polypeptides, aromatic po1yamidesl3)exhibit a persistence flexibility mechanism. At the same time, there is no fundamental reason why liquid-crystalline solutions of rotational-isomericsemiflexible macromolecules should not exist. Taking into account the recent intensive increase of the number of papers on the synthesis of new classes of stiff-chain polymers14 and the prevalence of the rota0 1984 American Chemical Society
